Chelsea went ahead through Salomon Kalou after a dreadful error by Everton goalkeeper Jan Mucha, but were pegged back by Louis Saha after goalkeeper Ross Turnbull was sent off.
But Everton wasted numerous chances to win it in normal time at Goodison Park and, after Royston Drenthe also saw red in extra time, Sturridge settled the fourth-round tie after 116 minutes.
It was a remarkable contest in which both sides missed a penalty, Nicolas Anelka firing wide early on and Petr Cech, on after Turnbull's indiscretion, saving from Leighton Baines.
Chelsea made 10 changes as they fielded a largely second-string team, but manager Andre Villas-Boas packed his bench with senior players as insurance.
Cech may have been the least likely of them to play, but he ended up contributing more than most as the Londoners reached the last eight.
While the likes of Frank Lampard, Fernando Torres and Juan Mata were also on the bench, captain John Terry did not travel.
Terry is currently the subject of a racism investigation but his absence was down to the fact he has been booked four times this season. A fifth yellow card at Goodison would have put him out of the weekend clash with Arsenal.
Everton fielded a side that would not have looked out of place in the Premier League, aside from the inclusion of regular cup goalkeeper Mucha in place of number one Tim Howard.
Chelsea went close to taking the lead in the first minute when Romelu Lukaku broke into the box and fired just wide.
Everton responded quickly as Diniyar Bilyaletdinov pulled the ball back for Saha but Turnbull just got down quick enough to block.
Saha then made another good run and tried his luck from 25 yards but Turnbull saved more impressively.
Chelsea should have taken the lead after John Heitinga conceded a penalty by hacking down Josh McEachran.
But Anelka lacked conviction as he plodded up to the spot and side-footed wide.
Chelsea wanted another penalty moments later when Anelka appeared to be bundled over by Sylvain Distin. It was possibly outside the box, but nothing was given anyway.
After Lukaku shot wide again, Everton briefly threatened when Drenthe reached the byline but Bilyaletdinov's shot from the edge of the area did not worry Turnbull.
Mucha, by contrast, was inexplicably troubled by an innocuous shot as Chelsea were gifted the lead after 38 minutes.
The Slovakian seemed in little danger as Kalou gently lobbed the ball at him from 18 yards, but he somehow allowed it to squirm through his grasp and his frantic effort to recover proved in vain.
Kalou almost seemed embarrassed to celebrate as he attempted to pull his shirt over his head.
Everton were almost caught napping early in the second half as Anelka found space only to shoot at Mucha, but Turnbull then suffered his moment of horror to allow Everton back into the game.
David Luiz failed to cut out Tim Cahill's flick and Saha instinctively touched the ball past Turnbull, who chopped him down.
Referee Lee Mason pointed to the spot and brandished a red card, but Cech immediately made amends for his team-mate with a brilliant double save.
The Czech, brought on as Lukaku was sacrificed, parried Baines' penalty and then blocked the left-back's second attempt before Cahill headed wide.
Cech saved again moments later from a Bilyaletdinov header and Everton suffered further frustration as Drenthe flashed a free-kick inches over.
Despite Cech's heroics, the tide had turned and Everton's pressure finally paid off as they equalised with seven minutes remaining when Saha met a cross from substitute Seamus Coleman with a perfect header at the near post.
Cech was well beaten, as he was moments later when Drenthe flashed a shot across the face of goal.
Everton maintained the tempo with another substitute, Denis Stracqualursi, heading over and Coleman making a mazy run into the area which Alex eventually cut out.
Saha saw another low shot roll agonisingly wide as Chelsea survived five additional minutes to reach extra time.
The break allowed Chelsea to regroup and the numbers were evened up when Drenthe was dismissed for a second bookable offence after sending Ryan Bertrand sprawling.
Anelka's misery continued as his fierce shot struck Baines on the line, hit the post and rebounded to safety, but Chelsea were not to be denied.
The winner came five minutes from the end as Mucha beat out a Florent Malouda shot but could do little to prevent Sturridge following in.